Yesterday afternoon there was an event at CUNY featuring a panel discussion on Chern-Simons terms. Nothing new there, although it was interesting to hear first-hand from Witten the story of how he came up with the Chern-Simons-Witten theory. One piece of news I heard from Nikita Nekrasov was that he was missing a talk that day at the Simons Center in Stony Brook by Maxim Kontsevich, who would be arguing that the Hodge and Tate conjectures were not true. The video of that talk has now appeared, see here.
I’m way behind in preparing for my class for tomorrow, so haven’t had time to watch the full video and ask experts about it. Will try and learn more tomorrow after my class, but it does seem that if Kontsevich is right that would be a dramatic development. If you are able to evaluate Kontsevich’s arguments, any comments welcome. Tomorrow I’ll also try and at least find some good references to suggest for anyone who wants to learn the background of what these conjectures say.
Update: I see there’s an older version of this idea described here.
Update: Having watched the video and talked to a few people about it, I fear that there’s not much new here, and nothing likely to convince experts that a falsification of the Hodge or Tate conjectures is on the horizon. Kontsevich himself introduces the talk as “not really a talk, but a kind of after-dinner rant.” He for a long time has been trying to find examples that could falsify the Hodge conjecture, with no success so far, and from what I can tell, he doesn’t have a new compelling proposal for where to look and how to do this.